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Homework Help: Proof by induction an expression

  1. Apr 28, 2012 #1
    I'm trying to prove by induction the expression:


    For the base case, n=2, S(2)=[itex]\frac{2(2-1)}{2}=1[/itex]

    For S(n+1)=[itex]\frac{(n+1)((n+1)-1)}{2}[/itex] I have:

    S(n+1) = [itex]\frac{n(n-1)}{2}[/itex] + (n+1) <--- Is this correct?

    I don't know what is the term for n+1. Any help?
    Last edited: Apr 28, 2012
  2. jcsd
  3. Apr 28, 2012 #2


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    Science Advisor

    There appears to be something missing here! What is being summed?

    No one can tell you until you tell us what is being summed.
  4. Apr 28, 2012 #3
    I corrected in the first message now.
  5. Apr 28, 2012 #4
    I solved. Here's the solution. Can you check it if it's right?

    For n=2, I got

    For n=3,

    So, for n, I got

    For n+1 I got

    For the LHS


    For the RHS

    So, the LHS = RHS, proofing that [itex]\frac{n(n-1)}{2}[/itex] is correct for any n.

    Is this solution correct?
  6. Apr 28, 2012 #5


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    What you have written is basically a "proof by induction" and, yes, it is correct.

    Actually, the first thing I would do is write that as
    [tex]\sum_{i= 1}^{n- 1} n- 1= \sum_{j= 0}^n j[/tex]
    where I have taken j= i- 1.

    If I really didn't want to do that, I might note that
    [tex]\sum_{i=1}^{n-1}= n\sum_{i= 1}^n 1- \sum_{i=1}^{n- 1}i[/tex]
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